Standing waves and global existence for nonlinear wave equations with potential, strong, and nonlinear damping terms
© Huang; licensee Springer 2014
Received: 27 October 2013
Accepted: 27 May 2014
Published: 10 September 2014
This paper is concerned with the Cauchy problem of nonlinear wave equations with potential, strong, and nonlinear damping terms. Firstly, by using variational calculus and compactness lemma, the existence of standing waves of the ground states is obtained. Then the instability of the standing wave is shown by applying potential-well arguments and concavity methods. Finally, we show how small the initial data are for the global solutions to exist.
For the case of linear damping (, ) and nonlinear sources, Levine  showed that the solutions to (1.1) with negative initial energy blow-up for the abstract version. For the nonlinear damping and source terms (, , , ), the abstract version has been considered by many researchers –. For instance, Georgiev and Todorova  prove that if , a global weak solution exists for any initial data, while if the solution blows up in finite time when the initial energy is sufficiently negative. In , Todorova considers the additional restriction on . Ikehata  considers the solutions of (1.1) with small positive initial energy, using the so-called ‘potential-well’ theory. The case of strong damping (, , ) and nonlinear source terms () has been studied by Gazzola and Squassina in . They prove the global existence of solutions with initial data in the potential well and show that every global solution is uniformly bounded in the natural phase space. Moreover, they prove finite time blow-up for solutions with high energy initial data. However, they do not consider the case of a nonlinear damping term (, , ).
For simplicity, throughout this paper we denote by and arbitrary positive constants by .
Preliminaries and statement of main results
From Proposition 2.1, it follows that is the critical case, namely for , a weak solution exists globally in time for any compactly supported initial data; while for , blow-up of the solution to the Cauchy problem (1.1) occurs.
The main results of this paper are the following.
There exists such that
(a2) is a ground state solution of (1.4).
From Theorem 2.3, we have the following.
(b1) If, and there existssuch that, then the solutionof the Cauchy problem (1.1) blows up in a finite time.
Variational characterization of the ground state
In this section, we prove Theorem 2.3.
This completes the proof of Lemma 3.1. □
is bounded below on M and.
Next, we solve the variational problem (2.6).
We first give a compactness lemma in .
Letwhenandwhen. Then the embeddingis compact.
In the following, we prove Theorem 2.3.
Proof of Theorem 2.3
Next we prove (a2) of Theorem 2.3.
So far, we have completed the proof of Theorem 2.3. □
Blow-up and global existence
In this section, we prove Theorem 2.5.
Let, thenandare invariant under the flow generated by (1.1).
Similarly, we can show that is also invariant under the flow generated by (1.1). This completes the proof of Lemma 4.1. □
Next, we prove Theorem 2.5.
Proof of Theorem 2.5
By (2.7), we have .
Next, we prove (b2) of Theorem 2.5.
Thus, we complete the proof of Theorem 2.5. □
- Gazzola F, Squassina M: Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2006, 23: 185-207. 10.1016/j.anihpc.2005.02.007MathSciNetView ArticleGoogle Scholar
- Soffer A, Weinstein MI: Resonance radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 1999, 136: 9-74. 10.1007/s002220050303MathSciNetView ArticleGoogle Scholar
- Levine HA:Instability and nonexistence of global solutions to nonlinear wave equations of the form . Trans. Am. Math. Soc. 1974, 192: 1-21.Google Scholar
- Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 1994, 109: 295-308. 10.1006/jdeq.1994.1051MathSciNetView ArticleGoogle Scholar
- Todorova G: Cauchy problems for a nonlinear wave equations with nonlinear damping term. Nonlinear Anal. 2000, 41: 891-905. 10.1016/S0362-546X(98)00317-4MathSciNetView ArticleGoogle Scholar
- Ikehata R: Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal. 1996, 27: 1165-1175. 10.1016/0362-546X(95)00119-GMathSciNetView ArticleGoogle Scholar
- Todorova G: Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. C. R. Acad. Sci., Sér. 1 Math. 1998, 326: 191-196.MathSciNetGoogle Scholar
- Zhang J: Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Z. Angew. Math. Phys. 2000, 51: 498-503. 10.1007/PL00001512MathSciNetView ArticleGoogle Scholar
- Huang WY, Lai SY, Zhang J: Sharp conditions of global existence for the nonlinear Klein-Gordon equation. Acta Math. Sin. 2011, 54: 435-442.MathSciNetGoogle Scholar
- Pucci P, Serrin J: Global nonexistence for abstract evolution equations with positive initial energy. J. Differ. Equ. 1998, 150: 203-214. 10.1006/jdeq.1998.3477MathSciNetView ArticleGoogle Scholar
- Liu WJ: Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms. Nonlinear Anal. 2010, 73: 244-255. 10.1016/j.na.2010.03.017MathSciNetView ArticleGoogle Scholar
- Zhang ZY, Miao XJ: Global existence and uniform decay for wave equation with dissipative term and boundary damping. Comput. Math. Appl. 2010, 59: 1003-1018. 10.1016/j.camwa.2009.09.008MathSciNetView ArticleGoogle Scholar
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